Ordinary Differential Equations and Limits (G12)

What is the integrating factor for the differential equation:  $\frac{1}{x}\frac{\text{d}y}{\text{d}x}-\frac{1}{1+x^2}y=x^3$  

•  $\ln\left(x^2\right)$  

•  $-\frac{1}{x^2}$  

•  $-\frac{1}{\sqrt{x}}$  

•  $\frac{1}{\sqrt{1+x^2}}$  

Which of the following mathematical models CANNOT be solved by using separation of variables

• $\frac{\text{d}P}{\text{dt}}=kP$

• $\frac{dT}{dt}=k\left(T-Tm\right)$

• $\frac{dA}{dt}=kA$

• $L\ \frac{dI}{dt}+RI=E\left(t\right)$

 Solve the given differential equations by separable variable method   $\frac{\text{d}v}{\text{d}t}=\frac{3+v^2}{v}$  

•  $v=\ln\left|3+v^2\right|+C$  

•  $v^2=e^{2t+2c}-3$  

•  $v=Ae^{2t}+3,\ \ \ \ \ A=e^{2c}$  

•  $v^2=Ae^{2t}+3,\ \ \ \ \ \ A=C$  

Please find the answer to this limit: $\lim_{x\rightarrow\infty}\frac{x^2+\sin x}{x^2}$  =

• 0

• 1

• -1

•  $\infty$  

Please find the solution of linear differential equation  $\frac{\text{d}y}{\text{d}x}+y=e^x$  

•  $y=\frac{e^2+c}{4x^2}$  

•  $y=\frac{ex^2}{4}+c$  

•  $y=\frac{e^{4x}+c}{x^2}$  

•  $y=\frac{e^x}{2}+Ce^{-x}$  

Please find  $\lim_{x\rightarrow0}\frac{2x+\tan3x}{\sin x}$  

• 0

• 2

• 5

•  $\infty$